Dynamics analysis of planar rigid-flexible coupling deployable solar array system with multiple revolute clearance joints

Mechanical Systems and Signal Processing 117 (2019) 188–209

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Dynamics analysis of planar rigid-flexible coupling deployable solar array system with multiple revolute clearance joints Yuanyuan Li, Cong Wang ⇑, Wenhu Huang School of Astronautics, Harbin Institute of Technology, Harbin, PR China

a r t i c l e

i n f o

Article history: Received 24 January 2018 Received in revised form 10 July 2018 Accepted 21 July 2018

Keywords: Rigid-flexible coupling Deployable solar arrays Clearance joint Dynamic response

a b s t r a c t This paper numerically analyzes action of journal at clearance joint and interaction of journal between multiple clearance joints to reveal the dynamic behavior of planar rigid-flexible coupling solar array system considering joint clearance in depth. A typical solar array model used is composed of a rigid main-body described by Nodal Coordinate Formulation (NCF) and two flexible panels described by Absolute Nodal Coordinate Formulation (ANCF). The system consists of two torque springs, one closed cable loop (CCL) configuration, two latch mechanisms and two clearance joints. The normal contact force effect and tangential friction effect at clearance joint are considered by using nonlinear contact force model and amendatory Coulomb friction model, respectively. Action and interaction of clearance joints are studied to indicate motion property of journal in initial phase, deployment phase and post-lock phase, which provide foundations for effect analysis of overall dynamic behavior. Then comparison results reveal the effects of joint clearance, panel flexibility and their coupling on dynamics of solar array system at these three phases. Coupled with clearance at collision phase, elastic vibration property of flexible panels dominates to cause system shock; while coupled with clearance at contact phase, suspension damping property of flexible panels dominates to steady the system. Finally, rational distribution of clearance size may provide a way to balance wear degree between joints. Decrease joint clearance with more intense collision could reduce the wear depth and balances wear degree between clearance joints. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Joint clearance is inevitable in articulated multi-body mechanisms due to the design, manufacture and assembly processes. Follows, a wear performance will intensify the clearance effect. Generally, vibration, noise, and large instantaneous impact forces at joints are the characteristics of the multi-body system with clearance joints. These joint clearance effects may reduce system’s accuracy, reliability and stability, lead to failure of the mechanism’s kinematic and dynamic outputs, or even destroy the mechanism. Over the last few decades, many researchers have studied effects of clearance on dynamic responses of planar and spatial multi-body mechanisms using theoretical and experimental approaches. Flores et al. [1–6] compared the kinematic and dynamic responses of rigid planar and spatial multi-body systems considering dry and lubricated friction, and studied the effects of clearance size and the operating conditions on predicting the dynamic responses of multibody systems. Erkaya ⇑ Corresponding author. E-mail addresses: [email protected] (C. Wang), [email protected] (W. Huang). https://doi.org/10.1016/j.ymssp.2018.07.037 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

Y. Li et al. / Mechanical Systems and Signal Processing 117 (2019) 188–209

189

Nomenclature T K l u / h A r
r CP S fp M M qe q

q Qex Q k e vt d n t d Rb Rj Ce Fn E u d_ ðÞ Ft cf cd

Uq k C Fc KW H S

p h

torque of mechanism (N
m) stiffness (N
m=rad) distance between two wheels (m) relative rotation angle of two wheels (m) angle between the center line of two wheels and the belt (m) angle (rad) rotation matrix position vector in global coordinate position vector in local coordinate coordinate transformation matrix shape function concentrated force mass matrix (kg) torque (N
m) generalized coordinates of element generalized coordinates of system density (kg/m3) generalized external force matrix generalized elastic force matrix curvature (m1) strain relative tangent velocity (m/s) eccentricity vector (m) normal direction tangential direction penetration depth (m) radius of the bearing (m) radius of the journal (m) restitution coefficient normal contact force (N) Young’s modulus Poisson’s ratio initial impact velocity tangential friction force (N) friction coefficient dynamic correction coefficient Jacobi matrix of constraint equation Lagrange multiplier column matrix damping matrix generalized contact force matrix dimensionless wear coefficient hardness of materials relative slippage distance of contact surface normal contact pressure wear depth

et al. [7–10] revealed that a flexible small-length flexural pivot plays suspension effects to decrease the undesired responses of the system with clearance joints by comparing the mechanism having rigid and flexible links. Tian Q et al. [11–15] used Absolute Coordinate Based method to establish flexible multibody systems with dry and ElastoHydroDynamic lubricated clearance joints. Marques et al. [16] presented a new formulation to model spatial revolute joints with radial and axial clearances. Zheng E et al. [17,18] used ADAMS software to model flexible multibody mechanism for ultra-precision presses and a closed high-speed press system. Salahshoor et al. [19] used multiple scales method to conduct a vibration analysis of a mechanical system with multiple clearance joints. And some researchers investigated kinematics and dynamics of 3-RRR and 4-RRR parallel mechanisms with clearance joints [20,21]. Marques et al. [25] used a spatial four-bar mechanism with a spherical joint as an application example to examine and quantify the effects of various friction force models, clearance sizes, and the friction coefficients. On the other hand for some biological applications, Askari et al. [22,23] developed a spatial multibody dynamic hip model using a friction-velocity constitutive law combined with a Hertzian contact model and

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investigated the influence of friction-induced vibration on the predicted wear of hard hip arthroplasties. Costa et al. [24] modeled a planar hip joint under the framework of multibody systems to compare the effects of the ideal, dry, and lubricated revolute joints utilized for the interaction of femur head. In addition, many researchers pay attentions to the contact models and friction models at clearance joints. Each wellknown friction model and contact force model has its own distinctive pros and cons by comparing their fundamental characteristics and their performances. Most existing viscoelastic contact force models were described and compared [26]. The comparative investigation shows that an appropriate selection of contact force model with reasonable dissipative damping is significant to analyze the dynamic response of mechanical systems [27]. Besides, different friction force models were investigated and their essential computational characteristics and physical properties were discussed and compared in detail [28–31]. Explicit criteria in the form of calculable upper bounds on the friction coefficients was provide to guarantee frictional contact problem remain well-posed [32]. From the comprehensive survey, it can be clearly observed that a large majority of the existing works have only been focused on the modeling and numerical algorithms for the simple mechanisms with clearance joints [33]. From research objective, most studies treated a four-bar mechanism [5,9] or a crank-slider mechanism with multiple clearance joints as an example to investigate the clearance effects [7–9,34–40]. The dynamic behaviors of a simple mechanism considering effects of joint clearance size, number of clearance joints, different clearance locations, contact stiffness, flexible components and so on were discussed adequately. Extension and application of the current methodologies on joint clearances to some complex application examples are needed to be considered, such as the space deployable structures [33]. However, a solar array system, which is a crucial component to provide indispensable power for the whole spacecraft, was often considered with perfect joints when most researchers modeled and controlled a spacecraft system. Fufa et al. [41] used ADAMS and ANSYS computer programs to simulate and model solar panels deployment and locking process, and the results demonstrated the effects on attitude of satellite. Wallrapp et al. [42] used the multi-body program SIMPACK to simulate the deployment process of a satellite solar array, and checked the influence of the flexibility of solar panels on the solar generator motions. Ding et al. [43] provided a root hinge drive assembly (RHDA) method to control the solar array deployment process and established a multi-DOF mechanism dynamics model to describe deployable solar array system. More recently, Liu et al. [44] proposed a rigid-flexible-thermal coupling dynamic model for satellite multibody system. Paolo et al. [45] analyzed the attitude behavior of the spacecraft with one flexible panel under the sloshing motion described through a spherical pendulum model. Alipour et al. [46] derived a precise compact dynamic model for an active stabilized spacecraft system with flexible members and then developed attitude control algorithm. Furthermore, Li et al. [47,48] presented a modeling method of using generalized variables to describe the joint coordinates to establish rigid solar array system with stick–slip friction at joints and designed an attitude controller to eliminate the drift of spacecraft using fuzzy adaptive PD control method. Although, recent years, several researchers pay attention to model solar array system considering joint clearance, these researches are still insufficient. Li et al. [49] and Zhang et al. [50] modeled solar array system with one revolute clearance joint and revealed that clearance joint brought evidently nonlinear dynamic characteristics to the solar array system deployment. However, modeling solar array system with just one clearance joint is not enough, while clearance number is a nonnegligible influence factor for the dynamic response of multi-body system. Furthermore, considering multiple clearance joints, Li et al. [51] used ADAMS software to analyze the effects of torsional spring, closed cable loops (CCL) configuration and latch mechanism on the dynamics of rigid solar array system, but the work ignored the flexibility of the panel. Although Li et al. [52] further given some dynamical phenomena of the rigid-flexible coupled solar array system with multiple clearance joints, but the work did not illuminate the mechanism of coupling effect and the cause of specific dynamic phenomena. Besides, these model used in previous researches has several drawbacks. The contact and friction models used under ADAMS software is not flexible enough and deep enough due to some parameters default setting. Such as penetration depth is set relying on operators’ experiences and doesn’t change in real time with computing process. And the flexible character of solar panel depends on modal truncation under finite element analysis software. However, the absolute nodal coordinate formulation has been developed for modeling dynamic systems of large-displacement and large-rotation problems in flexible multi-body systems, while the conventional finite element method deals with the small-displacement problems [11]. Combined with nodal coordinate formulation leads to a constant mass matrix for the rigid-flexible coupling multi-body system, and takes the vectors rather than rotational coordinates to describe the rotation and deformation of the rigid-flexible bodies, these contribute to the improvement of computation efficiency. Therefore, this paper selects Nodal Coordinate Formulation (NCF) and Absolute Nodal Coordinate Formulation (ANCF) combined to establish a typical rigid-flexible coupling solar array system with multiple clearance joints. The accurate modeling of spacecraft with large flexible appendages is the foundation for mechanism design, behavior prediction and control analysis. And the dynamic results are compared with the model established under co-simulation of ADAMS and finite element software in a later section. More importantly, this paper aims to study the journal action at a clearance joint and the interaction between the multiple clearance joints to explain the effect of joint clearance on dynamic response of the system in depth, and further to illuminate the mechanism of coupling effect of joint clearance and panel flexibility, that fundamentally reveals the reason for some specific dynamic performances of the rigid-flexible solar array system with multiple clearance joints. Compared to previous work, it’s no longer just presenting the dynamic phenomena. Besides, this paper considers inevitable wear at clearance joint and give some design suggestions for the solar array system with multiple clearance joints.

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The paper is organized as follows: Section 2 establishes the solar arrays model adopted in this paper and introduces the three key mechanisms of the model consists of torque springs, CCL and lock mechanisms. Section 3 describes formulations of NCF and ANCF briefly, gives contact model of planar revolute clearance joint under NCF -ANCF framework, and uses Archard’s wear model to describe wear calculation. Section 4 gives computational solutions to solve equations of motion of planar rigid-flexible multi-body system with clearance joints by using Generalized-a method. Section 5 gives calculation parameters of the solar array model. Section 6 shows the numerical results. These verify the validity of proposed model, analyze the action and interaction of clearance joints, reveal the effects of flexibility and clearance on the dynamic response, and discuss the effects of clearance distribution. Finally, the main conclusions are drawn in Section 7.

2. Description of deployable solar array system Deployable solar array system is the critical appendages of spacecraft. The essential and vital devices of the system are torsional spring mechanism (Fig. 1(a)), closed cable loop (CCL) mechanism (Fig. 1(b)) and latch mechanism (Fig. 1(c)). Once a satellite is launched into orbit, preloaded torsional springs provide driving force to deploy folded panels. Under the control of flexible CCL mechanisms, panels deploy synchronously. When panels are deployed to the same plane in space, latch mechanisms begin to work and latch the panels. A typical space deployable solar array adopted in this paper is shown in Fig. 1(d), which consists of one rigid main-body and two flexible panels connected by clearance revolute joints. Moreover, latch mechanisms and torsional springs are located in revolute joints. Furthermore, all the torques produced by these three key mechanisms on the adopted solar array system in this work are shown in Fig. 1(e), in particular two pairs of driving torques, a couple of equivalent synchronous torques and two groups of lock torques. The system is considered as a 2-dimensional case, all the forces and torques adopted in this paper are placed in a plane.

2.1. Torsional spring mechanism The driving torque generated from torsional spring mechanism on the i

th

(i = 1,2) joint can be expressed as [48,49]

T idriv e ¼ Kðhpre  hÞ

ð1Þ

where K is the torsion stiffness of torsion spring; hpre and h are the preload angle of torsion spring and practical deployment angle at the ith joint, respectively.

F2

F2

F1 (a) Torsional spring

Tccl

Fcclr

r2

F1

(b) Closed cable loops

B

C

r1

Fcclt

C

A

E A

D

Solar panels(n)

Joint (Trosion spring and latch mechanism) (d) Structure of solar array system

B F

D F ( ) Deployment phase ( ) Post-lock phase (c) Schematic diagram of the latch mechanism

CCL Spacecraft main-body

E

Tdrive2 Tdrive1 Spacecraft main-body

Tdrive1 1

Tdrive2

TCCL

2

Tlock1

Tlock2

TCCL

(e) Torque analysis diagram of the solar array model

Fig. 1. Space solar array model and its three key devices.

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2.2. CCL mechanism Fig. 1(b) illustrates a typical CCL mechanism, which is composed by synchronous wheels and pre-tensioned close-loop cables. The tight-side tension F1 and slack-side tension F2 can be expressed by




F 1 ¼ F 0 þ ðd  d0 ÞK ccl þ ur 1 K ccl

F1 P 0

F 2 ¼ F 0 þ ðd  d0 ÞK ccl  ur 1 K ccl

F2 P 0

ð2Þ

where F0 is the preload force of the belt; d is the real distance under tension and d0 is the initial distance between two wheels; Kccl represents the equivalent elastic stiffness; u is the relative rotation angle of two wheels; r1 and r2 are the radius of the wheels. When the deployment process of solar arrays is not synchronous, the difference value between the tight-side tension F1 and slack-side tension F2 will provide the radial force Fcclr, tangential force Fcclt and torque Tccl. So

8 > < F cclr ¼ ðF 1 þ F 2 Þ cos / F cclt ¼ ðF 1 þ F 2 Þ sin / > : T ccl ¼ ðF 1  F 2 Þr 1 ¼ 2ur 21 K ccl

ð3Þ

where / is the angle between belt and the center line of two wheels. CCL mechanisms supply a passive control torque in proportion to the angle difference to synchronize the deployment angles between solar panels. Thus, these passive synchronous torques T1 and T2 generated by the CCL can be regarded as

(

T 1 ¼ nur 21 K ccl ¼ K 1 ð2h1  h2 Þ T 2 ¼ ur 21 K ccl ¼ K 2 ð2h1  h2 Þ

ð4Þ

where K1 and K2 are the equivalent torsional stiffness of wheels; n is the ratio of the wheels; h1 and h2 are the practical deployment angles. 2.3. Latch mechanism Fig. 1(c) illustrates a typical latch mechanism. Two connected bodies A and B rotate round the joint C. Fig. 1(c) (I) shows that the pin D can move along the surface of E in the deployment process. When the deployment angle hi arrives the preset lock angle, the pin E slides into the groove F as shown in Fig. 1(c) (II). The latch mechanism is already activated and then the two connected bodies are latched at the expected position. Besides, the latch angle between spacecraft main-body and panel is 0.5p, the latch angle between panels is p. The equivalent latch torque Tlock adopted in this work is based on a STEP function and a BISTOP function. The STEP function uses a cubic polynomial to approximate the Heaviside step function increases from h1 to h2. The BISTOP function is a bilateral collision function between x3 and x4, at which point the collision function begins to push the pin back towards the expected center angle. The latch torque can be expressed as

8 if the mechanism is unlocked > > > > > T lock ¼ 0 > > > < > if the mechanism is locked > > > > > T lock ¼ STEPðhi ; x1 ; 0; x2 ; 1Þ > > : BISTOPðhi ; h_ i ; x3 ; x4 ; K bs ; e; C; dÞ

ð5Þ

STEPðhi ; x1 ; h1 ; x2 ; h2 Þ ¼ 8 if hi < x1 : 0 > > > > < if x1 6 hi 6 x2 :  2   hi x1 hi x1 > þ ðh  h Þ 3  2  h > 1 2 1 x x x x > 2 1 2 1 > : if hi > x2 : 1

ð6Þ

BISTOPðhi ; h_ i ; x3 ; x4 ; K bs ; e; C max ; dÞ ¼ 8 ifhi < x3 : > > > < MaxðK ðx  h Þe  h_ stepðh ; x  d; C ; x ; 0Þ; 0Þ 3 max 3 i i i bs 3 > if h > x : 4 i > > : MinðK bs ðhi  x4 Þe  h_ i stepðhi ; x4 ; 0; x4  d; C max Þ; 0Þ

ð7Þ

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where hi and h_ i are the deployment angle and the relevant rotation velocity at the ith joint, respectively; e is an exponent; Kbs and Cmax are the equivalent stiffness and damping coefficients of the latch mechanism, respectively; d is the distance depth. The specific parameters are set referencing literatures [47,48]. 3. Description of a planar revolute clearance joint based on NCF-ANCF In this work, the spacecraft main-body is described as rigid part by using NCF and two solar panels are described as flexible parts by using ANCF. Then a revolute clearance joint is established in the NCF-ANCF-based framework. 3.1. Formulations of NCF Fig. 2(a) shows a planar rigid body described by NCF element; its motion situation can be defined using two points and four global generalized coordinates. Thus, the rigid body coordinate can be expressed as

r j gT

qe ¼ f r i

ð8Þ

where ri and rj are the position vectors of the nodes i and j, respectively. The location of a generic point P of the element is defined by a position vector rp in the inertial system and
r p in the moving system, so that

rp ¼ ri þ A
r p ¼ r i þ c1 ðr j  r i Þ þ c2 n

ð9Þ

where A is the rotation matrix, c1 and c2 are the components of the vector
r p in the basis formed by the orthogonal vectors rj
Then  ri and n, the vector n follows the direction of Y.

(

rp ¼

xp

)

 ¼

yp

1  c1

c2

c1 c2




c2 1  c1 c2 c1

ri



rj

¼ C P qe

ð10Þ

where CP is a coordinate transformation matrix. The mass matrix can be established as

Z

M¼q

v

CT Cdv

ð11Þ

When a concentrated force fp is applied at a point P of an element, the generalized external forces can be established as

Q ex ¼ C Tp f p

ð12Þ

When a concentrated torque M is applied at an arbitrary point on the element, an equivalent pair of forces f and f can be used to instead of this torque. The pair of forces act on a plane perpendicular to the direction of M, which is applied at the beginning and end of a unit vector uf, as shown in Fig. 2(a). The unit vector here is defined by

uf ¼

ðr j  r i Þ  M jjðr j  r i Þ  Mjj

ð13Þ

and

f ¼ uf  M

ð14Þ

Therefore, the equivalent generalized force with respect to the natural coordinates can be established as

Y

f

i

rP

ri 2

-f rP

P

i

uf

M

ri Y

ri x

P

j

j

X

rj

ri1

r j1

rp

Y

k

rj

rj rj 2

X

ri1

Z

Planar NCF element

ri 2

ri

rj 2

a

i

b

r j1

x X

Planar ANCF deformable beam element

Fig. 2. Planar NCF and ANCF element.

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Q ex ¼ ðCTi f  CTiþuf f Þ

ð15Þ

3.2. Formulations of ANCF Fig. 2(b) describes the planar ANCF deformable beam element with two nodes. The position vector rp of an arbitrary point P acted on the two-dimensional beam element in the global coordinate system is represented by

" rp ¼

a0 þ a1 x þ a2 y þ a3 x3

#

b0 þ b1 x þ b2 y þ b3 x3

ð16Þ

¼ Sqe

where S is the element shape function and it can be obtained

   x2 x3 x2 x3  x2 x3 x x 2 x3 ;S3 ¼ 3 S ¼ ½ S1 I2 S2 I2 S3 I2 S4 I2 S1 ¼ 1  3 þ2 ;S2 ¼ l  2 þ 2 ;S4 ¼ l ð Þ þ l l l l l l l l l ð17Þ where I2 represents the identity matrix of size two and l denotes the element length. The eight global generalized coordinates of the deformable beam element can be expressed as

 T @r i1 @r i2 @r j1 @r j2 qe ¼ ½e1 ; e2 ; e3 ; e4 ; e5 ; e6 ; e7 ; e8 T ¼ ri1 ; ri2 ; ; ; rj1 ; rj2 ; ; @x @x @x @x

ð18Þ

where x denotes the nodal coordinates in element local coordinate system. The vector ½r k1 ; r k2 T (k = i, j) indicates the position k is tangent to the beam centerline. coordinates defined in the global coordinate system. The vector @r @x The mass matrix can be established as

Z

M¼q

ST Sdv

v

ð19Þ

If a concentrated torque s is applied at the first node i of a beam element, the generalized external torques can be established as

h Q ex ¼ 0 0 here f i ¼

se4 f 2i

se3 f 2i

0 0 0 0

i

ð20Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe3 Þ2 þ ðe4 Þ2 .

According to Euler-Bernoulli beams hypothesis, the generalized elastic force can be established as

Z Q ¼ 0

here f ¼

l

T

EAex0 S0 S0 dxqe 

1 f4

Z EIz 0

l

0T

0

S0 S0 dxqe

ð30Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1 l T 0T 0 q S S qe dx. l 0 e

3.3. Contact forces in revolute clearance joint This subsection establishes a mathematical model of revolute clearance joint to represent the contact forces in the equations of motion of rigid-flexible multi-body system. Fig. 3 gives the bearing and journal of a revolute clearance joint in the NCF-ANCF framework. Based on the NCF and ANCF, nodes i and j respectively indicate the center of the bearing and the center of the journal. The eccentricity vector d represent the difference between the node i and j, it can be expressed as

d ¼ rj  ri

ð31Þ

The vector n represents the normal direction of the collision surfaces between the bearing and the journal; accordingly, the vector t represents the tangential direction. Obviously, the vector n is the unit eccentricity vector. It can be expressed by

n¼

d jjdjj

ð32Þ

Then the penetration depth due to local deformation is evaluated as

d ¼ jjdjj  c

ð33Þ

where d is represented by the distance between P and Q in Fig. 5. c denotes radial clearance value, which is equal to Rb  Rj Here Rb and Rj respectively denote the radius of the bearing and the radius of the journal.

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Bearing

Mb i j

ri

Mj dF nj

t Fnb

P

rj rP rQ

Ftj

Journal

Ftb

Q n

Y

X Fig. 3. A revolute clearance joint in the NCF-ANCF framework.

Fig. 4. Computational scheme for the dynamic analysis.

16 clearance joint

12.5 0.0 -12.5 -25.0 2.00

5 crank rotations 15 crank rotations 25 crank rotations

ideal joint

Wear depth ( m)

3

Crank moment (10 Nm)

25.0

12 8 4 0

2.02

2.04 Time (s)

2.06

(a) Crank moment

2.08

0

60 120 180 240 300 Circumferential angle ( )

(b) Journal surface wear depth

Fig. 5. Comparison results of four bar mechanism obtained by the presented method.

360

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Point P is the contact point on bearing body and point Q is the contact point on journal body. Here are the position expressions of point P and Q respectively as

r P ¼ r i þ nRb ;

ð34Þ

r Q ¼ rj þ nRj

The velocity of these two contact points P and Q under the global coordinate system can be obtained by taking derivative of Eq. (34) as

_ b; r_ P ¼ r_ i þ nR _ j: r_ Q ¼ r_ j þ nR _

ð35Þ

_

r j r i d_ where n_ ¼ jjdjj ¼ jjdjj .

The normal velocity and tangential velocity need to be deduced to calculate the normal contact force and tangential friction force. Thus, the relative normal velocity can be expressed as

v n ¼ ½ðrP  rQ ÞT nn

ð36Þ

and the corresponding tangent velocity can be obtained by

v t ¼ ðr_ P  r_ Q Þ  v n

ð37Þ

In order to indicate the contact, impact and friction forces under the NCF and ANCF framework, the forces that acted on the contact points P and Q should be transformed into the forces that acted at the nodes i and j. In addition, the torques that applied on the bearing center node i and journal center node j can be obtained respectively by

M i ¼ F ti  Ri ; M j ¼ F tj  Rj

ð38Þ

Accordingly, normal contact force and tangential friction force can be obtained based on some appropriate contact and friction laws as generalized forces in the equations of motion of the solar array system. This paper selects Lankarani and Nikravesh model [53] to evaluate the normal contact forces, since it also considers energy dissipation in the contact-impact process. Besides, compared to other estimated contact force models, this model shows better ability to obtain accurate results [35]. It has been used for numerous studies [1,2,8,22–25] and has been validated by experimental studies [27]. The model is given as

"

FN ¼

#! 3ð1  C 2e Þd_ n Kd 1 þ 4d_ ðÞ n

ð39Þ

where FN is the normal contact force, K is the equivalent stiffness; d is the penetration depth of the contacting bodies; the

exponent n selects 1.5 for metallic surfaces; d_ and d_ ðÞ are the relative impact velocity and the initial impact velocity, respectively. Ce is the restitution coefficient; Here K depends on the material properties and the shape of the contacting surfaces, which is expressed as:

K¼



Rb Rj 3pðrb þ rj Þ Rb  Rj

rk ¼

4

1  u2k ðk ¼ b; jÞ pEk

12

ð40Þ

where Rb and Rj are the radii of the bearing and journal, respectively. Ek and uk are the Young’s Modulus and Poisson’s ratio associated with each component. Eq. (39) is used to simulate the normal contact forces because it accounts for energy dissipation and exhibits good numerical stability at low impact velocities. Moreover, Eq. (39) is valid for impact velocities lower than the propagation speed of elastic waves across the bodies. This criterion is fulfilled in the applications used in the present study. In particular, alternative normal contact force models have been developed over the last years, other contact models considering energy dissipation can be utilized instead. The interested readers can find relevant information on the contact models in the publications [26,27]. In general, Coulomb friction model is widely selected to calculate the friction force in contact-impact process. However, to prevent the numerical difficulties happen near the tangential velocity equal to zero, a modified Coulomb’s friction law is selected to represent the friction response in this work, which has been used for numerous studies [3,7,8]. Although LuGre model has the best accuracy for the micro stick-slip motion system by testing and comparing 5 friction models [31], the comparative investigation of effects of the LuGre model and the modified Coulomb model on the system dynamics shows that the moments obtained via these two friction models are very close [15]. Besides, the friction force model utilized does not significantly affect the dynamic response of the mechanism [25]. Although the selected modified Coulomb’s friction law cannot capture all the friction phenomena, it is enough here. Because this paper focuses on the dynamic analysis of the solar

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197

array system with clearance joints and the relatively simple model can enhance the computational efficiency. Certainly, other modified coulomb friction models eliminating the discontinuity can be utilized instead. The interested readers can find relevant information on the friction models in the publications [15,25,28–31]. Thus, the friction model selected here is denoted as,

F T ¼ cf cd F N

vt

ð41Þ

jjv t jj

where cf is the friction coefficient,

cd ¼

8 > <0

if kv t k 6 v 0

v 1 v 0 > : 1

if kv t k P v 1

kv t kv 0

if

vt is the relative tangential velocity. And dynamic correction coefficient cd is given by

v 0 6 kv t k 6 v 1

ð42Þ

where v0 and v1 are given tolerances for the tangential velocity. The parameters are adopted according to literature [15]. The dynamic correction factor cd can prevent friction force from direction changes at almost null values of the tangential velocity. 3.4. Wear depth of revolute joint In the case of the revolute clearance joint components, the wear would occur when bearing and journal are in contact and make relative motion. The most frequently used model is the Archard’s wear model [54], which is widely used by a great deal of works to predict wear pattern occurring in the clearance joint [55–57]. Thus, this paper selects the Archard’s wear model to calculate the wear of revolute clearance joint in solar array systems. The Archard’s wear calculation formula is described as,

V ¼ KW

FN S H

ð43Þ

where V is wear volume; KW is dimensionless wear coefficient; FN is external load; H hardness of materials; and S is relative slippage distance of contact surface. When the wear depth deserves more interest than the wear volume, Eq. (43) can be denoted as,

dh ¼

KW pdS H

ð44Þ

where h represents the wear depth and p indicates the normal contact pressure. The wear coefficient is adopted according to literature [58]. Wear depth given by Eq. (44) can be analyzed as an initial value problem using the Euler integration algorithm, the updating expression can be described as,

hiþ1 ¼ hi þ Dhi

ð45Þ th

th

which hi+1 is the total wear up to the i + 1 wear step, hi is the total wear up to the i wear step and Dhi is the amount of wear for the current wear step. 4. Computational solutions for dynamical equation Under the NCF-ANCF framework, the assembly of these planar rigid and flexible elements can be operated in accord with the traditional finite element method. Thus the nodal coordinates of each element qe can be transformed into the generalized coordinates of the whole rigid-flexible coupling multi-body system q. The dynamics equation of multi-body system is calculated using the Lagrange multiplier method. The equations of motion for a rigid-flexible coupling solar array system can be obtained in a compact form as

M

UTq

!

! Q e þ Fc € q ¼ _ q_  2Uqt  Utt ðUq qÞ k 0 q

UTq

ð46Þ

where M, C and K are the generalized mass matrix, damping matrix and stiffness matrix of the multi-body system, € is the generalized acceleration vectors matrix. Uq is the Jacobi matrix of constraint equation. k is the respectively. q Lagrange multiplier column matrix, Qe is the generalized force matrix and Fc is the generalized external contact force matrix. The relative motion states of journal and bearing are divided into three motions: free-flight movement, continuous contact and impact motion modes. In the contact and impact deformation phase, the intermittent collision and friction forces occur in the clearance joints between two colliding bodies and Fc is introduced into the equations of motion. In this paper, the Generalized-a method is applied to solve the systems of algebraic differential equations. Implicit solution is more advantageous to enhance calculating stability and avoids solving the complex generalized force Jacobin matrix. The number of unknowns of equations of motion Eq. (46) involved in the Generalized-a method reduces by half compared with that involved in Baumgarte Stabilization Method (BSM). Because the solution procedure of BSM needs to convert the n

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second-order differential equations of motion into 2n first-order differential equations. Besides, the selection of the Baumgarte parameters affects the violation of constraints and then affects the stabilization of solution process [59], Generalized-a method can avoid that. Classical dynamic equations of motion for constrained mechanical multi-body systems were compared in literature [60]. Energy-consistent simulation of frictional contact in rigid multibody systems was proposed based on implicit surfaces and penalty method [61], while the generalized-a scheme is preferred in this paper for its lower memory requirement and its ability to control the numerical damping. The calculation flow of the motion equations of solar array system with two revolute clearance joints is displayed in Fig. 4 and its procedures are listed in the following steps: (1) Build up the system code. Develop these matrices and vectors of each joint and component systematically using MATLAB code, and then transform them into overall matrices. Set the initial conditions for the system. (2) At time t, predict displacement coordinate qn+1 at the next moment tn+1 according to the roles of Generalized-a € nþ1 . Here Dt is the time method. Then give the corresponding velocity coordinate q_ nþ1 and acceleration coordinate q step. c and b are the algorithm parameters, and A-stability is guaranteed for 2b P c P 1=2. Here b = 1/4 and c = 1/2. (3) According to the system positions, conduct kinematic analysis on each revolute clearance joint concurrently and determine the penetration depths d using Eq. (33). (4) If the penetration depth d P 0, it means the contact-impact mode occurs. Compute the normal contact force (FN) using the Lankarani-Nikravesh contact force model given in Eq. (39) and the frictional force (FT) using the modified Coulomb friction law given in Eq. (41). Otherwise, make the normal contact and frictional forces equal to zero. (5) Transform the generated unilateral contact forces acted on the contact points to the center node of joints. Get friction moments applied at element nodes using Eq. (38). Then transfer these forces and resulting moments for each clearance joint to the generalized external contact force matrix Fc of the systems’ equations of motion. (6) Calculate driving torques Ts of torsional springs using Eq. (1), synchronization torques Tccl of CCL mechanisms using Eq. (4) and lock torque Tlock of latch mechanisms using Eq. (5), these are corresponding to qn+1. Then transfer these torques to the generalized external force matrix Qe of the systems’ equations of motion. (7) Solve the nonlinear equations of variables qn+1. The algebraic equations can alternatively be solved with qn+1 as pri€ nþ1 and q_ nþ1 in terms of qn, q_ n , q € n and q € nþ1 . Then obtain qnþ1 , q_ nþ1 and q € nþ1 . mary unknowns, by substituting q (8) Repeat steps 2 to 7 until the simulation time ends.

5. Numerical model and parameters A typical deployable solar array system composed of a rigid main-body and two flexible panels is modeled based on the NCF-ANCF to study the effects of multiple imperfect revolute joints and flexible components on its dynamic response. The simulation parameters of the system are listed in Table 1. Besides, the materials of journal and bearing are Steel and Brass, respectively. The radius of journal is set as rp = 0.01 m, and the radius of bearing is set as rs = 0.0102 m. In these simulations, the initial configuration of the mechanism is defined when the panels are folded without the angles of spread, and the centers of journal and bearing of the revolute clearance joint coincide. In addition, every flexible panel is divided into 6 elements, so the whole multi-body system has 60 degrees of freedom based on NCF-ANCF in this paper. To prove the validity of this model, a rigid-flexible coupling solar array model is established under co-simulation of Nastran and ADAMS software. Every flexible panel is established under Nastran and finite element model is set up with 60 beam elements.

Table 1 Parameters used in the dynamic simulation. Length /Width/Thickness of Main-body (m) Length of Panel 1(2) (m) Width of Panel 1(2) (m) Thickness of Panel 1(2) (m) Material Density of Main-body (kg/m3) Material Density of panels (kg/m3) Young’s Modulus of panels (Pa) Torsional stiffness of spring1 (N
m=rad) Torsional stiffness of spring2 (N
m=rad) Preloaded angle of spring1 (rad) Preloaded angle of spring2 (rad) Coefficients of restitution Ce Contact stiffness (K,N/m3/2) Friction coefficient cf Time step (s)

1 1.5 1 0.02 1500 200 280e6 0.1 0.075 0.5 p

p 0.9 7.39  1010 0.15 0.0001

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6. Results and discussion 6.1. Model verification

1.8

3.5

1.5

3.0

1.2 0.9 NCF ADAMS

0.6 0.3 0.0 0

5

10

15 20 Time (s)

25

30

Angular displacement of panel 2 (rad)

Angular displacement of panel 1 (rad)

An elementary four bar mechanism is established according to Ref. [55] as a demonstrative application to validate the multi-body dynamic model considering clearance joint and wear phenomenon under NCF-ANCF formulation used in this paper. Numerical simulations were performed under the MATLAB R2017a on a desktop computer with 3.3 GHz CPU and 8 GB RAM. When the time step is 0.000005 s for the total simulation setting 2.08 s, the model with ideal joint costs 22968 s and the model with clearance joint costs 25137 s to obtain the results. Fig. 5 gives the dynamic response of this typical case obtained by the method used in this paper. Fig. 5(a) shows the crank moment results of the model with ideal joint and with clearance joint. These two curves are almost identical with the crank moment results given by P. Flores in Ref. [55] (Fig. 7(c)). This illustrates the multi-body dynamic model with clearance joint used in this paper is correct. Besides, Fig. 5(b) gives the wear depth accumulated for 5, 15 and 25 full crank rotations plotted with the circumferential angle. The wear extent and wear trend are basically consistent with the wear results shown in Ref. [55] (Fig. 8)). The agreement proves the availability of wear prediction of the multi-body dynamic model used in this paper. Furthermore, the typical solar array system with ideal joints is used to prove validity of the dynamic model based on NCFANCF presented in this paper. To verify the validity of the rigid system described by NCF, the contrast model is established using ADAMS software; and to verify the validity of the rigid-flexible coupling system described by NCF-ANCF, the contrast model is established through co-simulation of Nastran-ADAMS software. Fig. 6 shows the comparison results of angular displacement of the rigid solar array system obtained by NCF method and ADAMS software and Fig. 7 shows the comparison results of angular displacement of the rigid-flexible coupling solar array system obtained by NCF-ANCF method and Nastran-ADAMS software, respectively. Obviously, these dynamic models based on NCF and ANCF achieve basically the same results as simulation software, which proves the validity of the presented model. Furthermore, from partial enlarged detail in Fig. 7, the curve obtained by Patran-ADAMS software lost flexible panel vibration information at about 18 s. Obviously, the rigid-flexible coupling solar array model established by this paper shows more accurate and complete vibration information compared with co-simulation results obtained by Patran-ADAMS software. It demonstrates using NCF-ANCF formulation to model the rigid-flexible solar array system can obtain more accurate

2.5 2.0 1.5

NCF ADAMS

1.0 0.5 0.0 0

(a) Angular displacement of panel 1

5

10

15 20 Time (s)

25

30

(b) Angular displacement of panel 2

Fig. 6. Comparision results of angular displacement of the rigid solar array system.

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Fig. 7. Comparision results of angular displacement of the rigid-flexible coupling solar array system.

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Initial phase

0.2

Contact I

0.0 -0.1

Contact III

Contact II

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X(mm)

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0.0 0.1 0.2 X(mm) Contact I Initial phase

0

Contact III

0.0 -0.2

Post-lock phase

(a) Journal center locus

(b) Movement locus of X direction and Ydirection with time

Fig. 8. Journal center locus of Joint 1 in the rigid system.

dynamic response. Thus, the superiority of the presented method could be proved when applied to establish spacecraft model with large flexible attachment. Figs. 6 and 7 show the deployment process of the solar panels. Solar panels are deployed by driving torsional springs in orbit and the deployment angles are controlled by CCL mechanism under the proportionality coefficient 0.5. Then, the system is locked by the latch mechanism at approximately 13.1 s. Finally, the system steadily keeps plane deploy state with the deployment angle of Panel 1 almost 0.5 p and the deployment angle of panel 2 almost p. 6.2. Action and interaction inside the clearance joints The simulation results of rigid solar array system with clearance joints show the dynamic action of every clearance joint and interaction of these two clearance joints, which provides foundations to explain the effect of clearance joints on the overall dynamic behavior of the whole system in the continue subsection. Figs. 8 and 9 show the center locus of journal of the two clearance joints in the rigid solar array system, respectively. Hinge pin center locus denotes the relative position between the center of journal and the center of bearing at the clearance joint. The locus runs limited in a circular area depending on the size of clearance. Movement locus of X direction and Y direction give the information of center locus of journal with time, which helps to explain mechanical behavior over time. Joint 1 is the joint between satellite and panel, and Joint 2 is the joint between two panels. As can be seen from Figs. 8 and 9, the whole motion state of solar array system can be divided into three phases: initial phase, deployment phase and post-lock phase. (1) Initial phase. This phase presents unilateral collision property as shown in Figs. 8(a) and 9(a) and happens at the beginning of the movement, about 0–0.950 s for Joint1 (Fig. 9(b)) and about 0–0.972 s for Joint2 (Fig. 9(b)). Folded solar array system is released and driven by torsional springs as the system startup operation. At that moment, the

Initial phase

Deployment phase Contact I

0.1

X(mm)

Y(mm)

0.2

0.0

-0.1

Y(mm)

Contact II

Contact III

Contact III

0.2 0.0 -0.2

-0.2 -0.2 -0.1 0.0 0.1 0.2 X(mm) Contact I Initial phase

Contact II

Post-lock phase

0

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15 20 Time (s)

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Post-lock phase

(a) Journal center locus

(b) Movement locus of X direction and Ydirection with time

Fig. 9. Journal center locus of Joint 2 in the rigid system.

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driving force make journal hit on the inner wall of bearing. And journal remains continuous and amplitude attenuated impact until it decays to almost zero. The system moves to the next phase. The impact force brings certain disturbance to the whole solar array system during the initial phase. (2) Deployment phase. This phase presents continuous contact property and exists in the stable deploy stage about 0.950–13.175 s for Joint1 as shown in Fig. 8 (b) and about 0.972–13.183 s for Joint 2 as shown in Fig. 9(b). Journal keeps almost creep against the inner wall of bearing throughout this phase. Thus, at this phase, dynamic performance of the whole system closes to the system with ideal joints and the contact force stays around some small value. That means the rigid solar array system is basically not affected by joint clearance during the deployment phase. Moreover, the deployment phase can be further detailed into three contact phases: Contact I phase, Contact II phase and Contact III phase, as shown in Figs. 8 and 9 Contact I phase: journal climbs along inner wall of bearing over and over again until the journal reaches the peak of the inner wall in Y-direction at about 10 s; Contact II phase: journal moves down along inner wall of bearing from the peak until latch mechanism startup; Contact III phase: at the locked moment, journal slides fast along the inner wall making clockwise circular motions at Joint 1 and anticlockwise circular motions at Joint 2. So far, the system is locked. (3) Post-lock phase. This phase presents bilateral collision property and lasts for a long time after the system latch until the whole system is stable. And this phase may appear when the satellite is disturbed in the space. Affected by the bilateral lock torque, journal violent impacts on the inner wall of bearing as shown in Figs. 8 and 9, which generates great impact force. Thus, at this phase, joint clearance makes evident effect on the dynamic performance of the whole solar array system. However, the system would gradually stabilize as energy dissipation over time. In addition, by contrasting journal locus at Joint 1 and Joint 2, the motion states are basically synchronous between the joints installed in the system. Joint 2 follows Joint 1 to experience all these three phases successively due to synchronous control torques from CCL mechanism. Accordingly, Fig. 10 shows joint forces at these two clearance joints. At the initial phase, unilateral collision generates certain impact force; the stable deployment motion produces little impact forces; Then, drastic impact forces appear in the two joints at the moment of latch completion due to violent bilateral collision, and the uninterrupted instantaneous impact force would last and gradually reduce until the system reaches steady state. In terms of the effects on the dynamics of solar array system, the post-lock phase deserves more attention. Because the degree of impact is greater and the frequency of impact is denser at this phase. From Fig. 10, it is obvious that the occurrence time of drastic impact forces between Joint 1 and Joint 2 are consistent, at about 15.74 s, 17.63 s, 20.59 s, 28.48 s and 29.33 s, which keep pace with the occurrence time of significant lock torque as shown in Fig. 11. It indicates that the lock torque from latch mechanism directly causes and affects impact force at clearance joints. Similarly, synchronous control torques curves in Fig. 12 also show the same performance. Inflection points of CCL torque curves, which do not vary with the vibration as result of flexibility of belt and wheels, represent the occurrence time of lock torque. Thus, the CCL torque and lock torque curves of solar array system could provide prediction of contact force curve at clearance joints. Furthermore, the contact performance can be optimized to improve the dynamics of solar array system after latch through adjusting design parameters of CCL mechanism and latch mechanism. Literature [53] presented significant design guidance for the key parameters of these mechanisms in details.

6.3. Effect of joint clearance and panel flexibility

800 700 600 500 400 300 200 100 0

600

Contact force (N)

Contact force (N)

The simulation results of rigid-flexible coupling solar array system with clearance joints show the performance of clearance joints and reveal effects of clearance, flexibility and their coupling on the overall dynamic behavior of solar array system.

500 400 300 200 100 0

0

5

10

15 Time (s)

20

25

(a) Contact force at Joint1

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0

5

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15 Time (s)

20

(b) Contact force at Joint2

Fig. 10. Contact force of rigid system.

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Lock torque (Nm)

30

Lock 1 Lock 2 20

10

0

10

15

20 Time (s)

25

30

CCL torque (Nm)

Fig. 11. Lock torque from latch mechanism of rigid system.

3 2 1 0 -1 -2 -3 -4

CCL1 CCL2

0

5

10

15

20

25

30

Time (s) Fig. 12. Synchronous control torques from CCL mechanism of rigid system.

Figs. 13 and 14 show the journal center locus of the two clearance joints in the rigid-flexible coupling solar array system, respectively. The system also experiences three phases consisting of initial phase, deployment phase and post-lock phase. Compared with Figs. 8 and 9, Figs. 13 and 14 show the effects of panel flexibility on center locus of journal at these three phases. (1) Initial phase. Disturbance lasts about 0–1.143 s for Joint1 and about 0–1.34 s for Joint2. The duration of initial phase of rigid-flexible coupled system is longer than rigid system. (2) Deployment phase. Although overall motion state is continuous contact, unilateral collision also happens in this phase, especially when journal climbs near the peak of inner wall of bearing. That means the smooth deploy process for rigid system is no longer so much stable for rigid-flexible coupled system. Besides, journal do not make circular motion along the inner wall at contact III phase. (3) Post-lock phase. Obviously, this phase presents more drastic bilateral collision performance. The collision frequency is significantly denser and the impact trace becomes more disordered. Consequently, disturbance produced by clearance joints of rigid-flexible coupling solar array system is more violent than that of rigid system after latch. The panel flexibility causes these differences between performance of clearance joints of rigid system and rigid-flexible system. Because the elastic vibrations of large flexible panels produce unceasing deformation displacement to make journals and bearings collide with each other continually, in particular the panels are constantly loaded bilateral collision forces. Accordingly, Fig. 15 shows joint forces at the two clearance joints of rigid-flexible coupling system. Compared with rigid system (Fig. 10), the impact force frequency become considerably higher, but impact force value become smaller. Similarly, the denser frequency is due to the uninterrupted elastic vibration of large flexible panels. However, in addition to elastic vibration, flexibility of panel also brings elastic damping to solar array system. Hence, the flexible panels as suspension damper reduce the impact force between journals and bearings at clearance joints. Moreover, because of the flexibility of panels, lock torque frequency of rigid-flexible coupling system become more intensive and the maximum of lock torque at lath moment reduced, as shown in Fig. 16. Fig. 17 shows CCL synchronous control torques of rigid-flexible coupling system contrasted by that of rigid system in Fig. 12. Obviously, amplitude of equivalent synchronous torque of flexible solar arrays model increases drastically, which is proportional to the difference of deploy angles according to Eq. (4). The large elastic deformation of flexible panels coupled to the rigid motion enlarge the difference and cause high fluctuation of CCL torque. Besides, the elastic deformation of flexible panels brings much tremble of CCL torque. These forces and torques acted in solar array system make evident effects on the dynamic performance of the system. Four comparison models are established to study effects of joint clearance and panel flexibility on dynamic response of solar

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Deployment phase

Initial phase

Post-lock phase

0.2

Contact I

X(mm)

Y(mm)

0.1 0.0 -0.1

Contact III

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0.0 0.1 0.2 X(mm) Contact I Initial phase

Contact II

0.2 0.0 -0.2

Post-lock phase

(a) Journal center locus

(b) Movement locus of X direction and Ydirection with time

Fig. 13. Journal center locus of Joint 1 in the rigid-flexible coupling system.

Initial phase

Deployment phase

Post-lock phase

0.2

Contact III

Contact II

0.2 0.0

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Contact III

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Post-lock phase

(a) Journal center locus

(b) Movement locus of X direction and Ydirection with time

Fig. 14. Journal center locus of Joint 2 in the rigid-flexible coupling system.

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Contact force (N)

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150 100 50 0

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0

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15 20 Time (s)

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(a) Contact force at Joint1

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0

5

10

15 20 Time (s)

(b) Contact force at Joint2

Fig. 15. Contact force of rigid-flexible coupling system.

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Lock torque (Nm)

30 Lock 1 Lock 2 20

10

0 10

15

20 Time (s)

25

30

CCL torque (Nm)

Fig. 16. Lock torque from latch mechanism of rigid-flexible coupling system.

4 3 2 1 0 -1 -2 -3 -4

CCL1 CCL2

0

5

10

15 Time (s)

20

25

30

Fig. 17. Synchronous control torques from CCL mechanism of rigid-flexible coupling system.

array system, consisting of rigid system with ideal joints (rigid-ideal model), rigid system with clearance joints (rigidclearance model), rigid-flexible coupled system with ideal joints (flexible-ideal model) and rigid-flexible coupled system with clearance joints (flexible-clearance model). Results taking panel 1 as an example, Figs. 18 and 19 show the comparison of angular displacement and angular velocity, respectively. The dynamic response of solar array system is closely related to the journal action of clearance joint at the three phases. And joint clearance, panel flexibility and their coupling make remarkable effects on the system at initial phase and post-lock phase. To analyze the effects in detail: (1) Initial phase. The black line in Fig. 18(b) shows the relative angular displacement between the rigid system with clearance joint and without clearance joint, it can be seen that the impact at clearance joint bring small jitter to the solar array system at this start-up stage. This phenomenon can be clearly observed in the partial enlarged drawing at initial phase in Fig. 19(c), comparing the angular velocity results of rigid system with clearance joints and the system with ideal joints. Then the blue line in Fig. 18(b) shows the relative angular displacement between rigid-flexible system and rigid system with ideal joints, it reveals that considering panel flexibility make the system shock due to the elastic vibration of panels. The angler velocity results in Fig. 19(d) visually shows the vibration of the rigid-flexible system compared to the rigid system. Furthermore, the coupling effect of joint clearance and panel flexibility on the dynamics of system is enhanced obviously at this phase. The results of rigid-flexible coupling system with clearance joints show conspicuous concussion of panels as the red line shown in Figs. 18(b) and 19(a) at initial phase, because the coupling of impact force and elastic vibration arouses the shock of solar array system. (2) Deployment phase. Obviously, effects of joint clearance are negligible at his phase according to locus analysis of journal in Section 6.2. Compared with the rigid-ideal model, the angular displacement of rigid-clearance model appears just a little fluctuation, as black line shown in Fig. 18(c). While elastic vibration of flexible panels continues to this phase, as the results of rigid-flexible coupling system with ideal joints shown in Figs. 18(c) and 19(d). However, compared with the results of rigid-flexible coupling system with ideal joints, angular displacement of rigid-flexible coupling system with clearance joints shows more stable performance at this phase (see red line in Fig. 18(c)). It means the coupling effect of clearance joint and panel flexibility reduces the vibration of the system at deployment phase. The angular velocity of flexible-clearance model obviously show this weakening phenomenon comparing with the result of flexible-ideal model, as shown in Fig. 19(b). Because the impact force generated by contact motion is nearly to zero as shown in Fig. 15, suspension damping action of clearance joint plays a dominant role in the coupling effects of joint clearance and panel flexibility at this deployment phase.

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Relative angular displacement (rad)

Post-lock phase

1.5 1.0

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.5 0.0 0

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10 15 Time (s)

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(a) Angular displacement of panel 1

0.0006

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1.56 15

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(c) Relative angular displacement at development phase

rigid-clearance flexible-clearance

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25

30

Time (s)

(d) Partial enlarged detail at post-lock phase

Fig. 18. Comparison of angular displacement responses.

(3) Post-lock phase. The partial enlarged detail of this phase in Fig. 19(c) shows the effect of clearance joint on angular velocity of the panel, the impact force make the system appears some wobble following with the latch force. More significantly, the oscillation of the system with clearance joint always falls behind the system with ideal joint, as shown in Fig. 18(d). It means that the existence of clearance joint brings vibration hysteresis quality to the solar array system. The clearance at joints may reduce the adjustment sensitivity of CCL mechanism due to the diminished change rate of develop angles. Consequently, the vibration frequency of synchronous control torques from CCL mechanism decline, which reduces the vibration frequency of the whole solar array system. Then Fig. 18(d) shows that the panel flexibility make the system become more significant fluctuate at post-lock phase, comparing the angular displacement of rigid-flexible coupling system and the rigid system, due to the changes of CCL torque of rigid system and rigidflexible system. Besides, the difference of development angles due to elastic vibration of flexible panel brings burrs to the fluctuation. Furthermore, affected by coupling of joint clearance and panel flexibility, the greater impact force and denser impact frequency make angular velocity of rigid-flexible coupling system with clearance joints shock dramatically at post-lock phase, as shown in Fig. 19(a). Similarly, because the coupling of drastic bilateral impact force and elastic vibration causes the violent vibration of solar panels. However, the suspension damper action of rigidflexible system with clearance joints could reduce the burr phenomena, as shown in Fig. 18(d). In general, joint clearance, panel flexibility and their coupling have negative effects on the dynamic performance of solar array system. 6.4. Effects of distribution of clearance size Literature [51] discussed effects of mechanism parameters on dynamics of solar array system with clearance joints and provided some optimization criterions from macroscopic perspective. This paper aims to study effects of joint clearance on the system whose mechanism parameters have been designed and determined. Thus, this section discusses the effects of distribution of clearance size on the dynamics of the system from clearance action perspective, in small. From Fig. 9, it can be seen that journal center locus of Joint 2 is more chaotic and more intensive than that of Joint 1. Because Panel 1 reaches to the latched position ahead of the Panel 2 under the driving torques of determined torsional springs. Panel 2 must be adjusted to keep synchronization with Panel 1 by the CCL mechanism. Consequently, Panel 2 appears more obvious oscillation characteristics than Panel 1 after lock, as shown in Fig. 7. For the same reason, Fig. 11 shows that the lock torque generated by latch mechanism at Joint 2 is greater and need more time to stabilize and dissipate.

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Angular velocity (rad/s)

Initial phase Deployment phase Post-lock phase

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rigid-ideal rigid-clearance

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(c) Effect of clearance on angular velocity

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Fig. 19. Comparison of angular velocity responses.

Contact force (N)

Contact force (N)

Two rigid models with different clearance distributions are established to discuss the effects of distribution of clearance size. Model 1 with 0.1 mm clearance at Joint 1 and 0.2 mm clearance at Joint 2, Model 2 with 0.2 mm clearance at Joint 1 and 0.1 mm clearance at Joint 2. Results of contact force of these two models are shown in Figs. 20 and 21. Comparison results exhibit that frequency of contact force generated at the joint with 0.1 mm clearance is denser than that at the joint with 0.2 mm clearance. Because smaller clearance shortens the collision distance between journal and inner wall of bearing. Collision frequency increases and impact density improves. However, the value of contact force generated at the joint with 0.2 mm clearance is larger than that at the joint with 0.1 mm clearance. Because the larger clearance lead to the greater contact stiffness and the higher impact velocity. Collision strength increases at the larger clearance joints. To compare the wear degree at clearance joint with different size, Figs. 22 and 23 gives the wear depth of these two models. Obviously, the wear depths of Model 2 are

joint 1

300 200 100 0

15

20 Time (s)

25

30

joint 2

300 200 100 0

15

20 Time (s)

25

30

Fig. 20. Contact force of Model 1 (joint1 clearance = 0.1 mm and joint2 clearance = 0.2 mm).

Contact force (N)

Y. Li et al. / Mechanical Systems and Signal Processing 117 (2019) 188–209

300

207

joint 1

200 100 0

15

20

25

30

Contact force (N)

Time (s) 300

joint 2

200 100 0

15

20 Time (s)

25

30

Fig. 21. Contact force of Model 2 (joint1 clearance = 0.2 mm and joint2 clearance = 0.1 mm).

7.0x10

-14

joint 1 joint 2

Wear depth (m)

-14

6.0x10 -14 5.0x10 -14

4.0x10 -14 3.0x10 -14

2.0x10 -14 1.0x10 0.0 0

60 120 180 240 300 Circumferential Angle (°)

360

Fig. 22. Wear depth of Model 1 (joint1 clearance = 0.1 mm and joint2 clearance = 0.2 mm).

7.0x10

-14

joint 1 joint 2

Wear depth (m)

-14

6.0x10 -14 5.0x10 -14

4.0x10 -14 3.0x10 -14

2.0x10 -14 1.0x10 0.0 0

60 120 180 240 300 Circumferential Angle (°)

360

Fig. 23. Wear depth of Model 2 (joint1 clearance = 0.2 mm and joint2 clearance = 0.1 mm).

much smaller and the wear degrees between multiple clearance joints are more balanced. While the wear depth of joint2 with 0.2 mm clearance of Model 1 increases evidently. It means that to reduce clearance size of the joint where collision is more intense could reduce the wear depth and balance wear degree between clearance joints; and to reduce clearance size of the joint where collision is slighter could intensify collision degree at the joint, which is already serious impacted. The more intense collision includes the larger value of impact force and the denser frequency of impact locus. Joint2 in this paper is the joint with more intense collision, as shown in Figs. 8–10. Thus, rational distribution of clearance size may be a good way to improve dynamic performance of solar array system, which avoid system breakdown due to over wear of some joint and prolong the service life. 7. Conclusions Rigid-flexible solar array system with clearance joints is modeled to study the dynamic action and interaction inside the clearance joints and to reveal effects of clearance, flexibility and the coupling on the dynamic performance of deployment of solar arrays system. The deployable solar arrays system consists of a rigid main-body described by NCF method and two

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flexible panels described by ANCF method. The nonlinear Lankarani and Nikravesh model and the amendatory Coulomb friction model are applied to evaluate the contact-impact effect and the friction effect at clearance joint. The wear model is established using Archard’s wear model. In addition, rigid solar array model is verified by using ADAMS software and rigid-flexible solar array model is verified by co-simulation of Nastran and ADAMS software. Firstly, dynamic action of every clearance joint and interaction of the two clearance joints are discussed in three phases: initial phase, deployment phase and post-lock phase, according to the motion state of journal inside bearing. For the rigid system, initial phase presents unilateral collision property. Deployment phase presents continuous contact property and contact force is nearly to zero. Post-lock phase presents bilateral collision property and drastic collision lasts for a long time. Contrastively, these performances change affected by flexible panels. For the rigid flexible system, the collision frequency is significantly denser and the impact trace becomes more disordered at initial phase and in particular post-lock phase. Moreover, the interaction of the two clearance joints presents synchronous property of motion phase between the joints installed in the system. And contact force curve at clearance joints could be predicted by CCL torque and lock torque curves of solar array system due to the synergistic consistency. Then, the effects of joint clearance, panel flexibility and the coupling on dynamics of solar array system are studied in these three phases according to the action and interaction. (a) Joint clearance causes impact and brings certain disturbance to the solar array system at initial phase and post-lock phase. Besides, joint clearance decreases the control frequency of CCL mechanism and brings fluctuation hysteresis quality to the system after latch. (b) Elastic vibration coupled with overall motion, panel flexibility brings the obvious vibration into the solar array system, especially in post-lock phase. Because large elastic deformation causes high fluctuation of CCL torque after latch. (c) For rigid-flexible solar array system with multiple clearance joints, coupled with impact forces at collision phase, elastic vibration property of flexible panels dominates to cause system shock; while coupled with clearance at contact phase, suspension damping property of flexible panels dominate to steady the system. Finally, effects of clearance distribution on dynamics of the system is discussed. To reduce clearance of the joint where collision is more intense may be a good way to balance abrasion degree between joints and improve dynamic performance of solar array system. 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